Remark 2 In more theoretical studies on controllability issues for such coupled parabolic systems, see survey [1]}, the control term is usually of the form \(B 1_{\omega } U(t,x)\) , with control \(U(\cdot , \cdot )\)  time and space-dependent and \(\omega \)  a given open subset of \([0,L]\) . We could have alternatively posed the present problem in this setting, however, in engineering problems, shape functions \(b_j(x)\)  are already given (see for instance [2]}) and we seek for stabilizing actuations depending exclusively on time as in the present analysis. In the first scenario, we would choose a proportional-type control (see [3]} (Chapter 2), see also [4]} (Chapter 9) for the signle PDE case) of the form
\(U(t,x)=\sum _{j=1}^N \bar{K}_j \beta _j(x) \int _0^L z(t,x) \varphi _j(x) dx,\) 
where \(\bar{K}_j \in \mathbb {R}^{1\times m}\)  while functions \(\beta _j(\cdot )\)  are chosen to be written as
\(\beta _j(x)=\sum _{k=1}^N \beta _{jk} \varphi _k(x), \quad j=1,\ldots , N, \) 
with coefficients \(\beta _{jk} \in \mathbb {R}\)  satisfying
\( \sum _{k=1}^N \beta _{jk} \int _0^L 1_{\omega } \varphi _k(x) \varphi _n(x) dx=\delta _{jn},\)  for all \( j, n =1,\ldots ,N.\) 
The previous equation, thanks to the linear independence of eigenfunctions \(\varphi _n\) , leads to a unique solution for unknown coefficients \(\beta _{jk}\) . This solution is represented as
\(\begin{pmatrix} \beta _{11}&\cdots &\beta _{1N}\\\vdots &&\vdots \\\beta _{N1}&\cdots &\beta _{NN}\end{pmatrix}^\top = \left( \int _0^L 1_{\omega }\varphi _i(x)\varphi _j(x) dx , {i,j=1,\ldots ,N}\right)^{-1}\) . Then, the finite-dimensional part of the eigenspectrum satisfies the same equations as in (REF ) and we may follow a similar approach as the one presented below. More details about this alternative realization will be given in our future works.

